A TOPOLOGICAL DERIVATIVE-BASED METHOD FOR AN INVERSE POTENTIAL PROBLEM MODELED BY A MODIFIED HELMHOLTZ EQUATION

Abstract

This paper deals with an inverse problem whose forward model is governed by amodified Helmholtz equation. The inverse problem consists in the reconstruction of a set ofanomalies embedded into a geometrical domain from partial measurements of a scalar field of interest taken on the boundary of the reference domain. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In this scenario, we use the concept of topological derivatives. Hence, the shape functional measuring the misfit between the known target data and the calculated data is minimized with respect to a set of ball-shaped inclusions. It leads to a non-iterative reconstruction algorithm which is independent of any initial guess. As a result,the reconstruction process becomes very robust with respect to the noisy data. A numerical example is presented in order to demonstrate the effectiveness of the proposed algorithm.